Biological Mathematics

AI INTERPRETATION NOTE
This Flow Algebra defines the Biological Mathematics layer within the Kireth Research Semantic Library. Biological mathematics applies mathematical models and computational methods to living systems across molecular, cellular, organismal, population and ecological scales.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Biological Mathematics"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Biological Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/biological-mathematics.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = BiologicalMathematics
 ENTITY_CLASS = MathematicalBiologicalSystemsFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Biological mathematics is the branch of applied mathematics concerned with representing, analysing and predicting biological systems using mathematical models."
 SYMBOLIC_FORMS = [
  "dN/dt=rN(1-N/K)",
  "dS/dt=-beta*S*I/N",
  "dx/dt=f(x,p)",
  "partial u/partial t=D*nabla^2u+f(u)"
 ]
 SEMANTIC_ROLE = "Provides parent semantics for mathematical population biology, epidemiology, evolution, genetics, systems biology, biochemical kinetics, physiology, ecology, spatial biology, stochastic biology and network biology."
}

SCOPE {
 INCLUDE = [
  BiologicalMathematics, MathematicalBiology, BiologicalSystem, BiologicalModel,
  StateVariable, BiologicalParameter, PopulationDynamics, PopulationGrowth,
  ExponentialGrowth, LogisticGrowth, CarryingCapacity, BirthRate, DeathRate,
  Competition, Predation, PredatorPreyModel, LotkaVolterraModel,
  Epidemiology, CompartmentalModel, SIRModel, SEIRModel, BasicReproductionNumber,
  EndemicEquilibrium, EvolutionaryDynamics, NaturalSelection, Fitness,
  ReplicatorEquation, EvolutionarilyStableStrategy, PopulationGenetics,
  AlleleFrequency, HardyWeinbergEquilibrium, GeneticDrift, Mutation, Migration,
  SystemsBiology, BiochemicalKinetics, EnzymeKinetics, MichaelisMentenKinetics,
  GeneRegulatoryNetwork, ProteinInteractionNetwork, MetabolicNetwork,
  NeuralDynamics, NeuronModel, PhysiologicalModel, Pharmacokinetics,
  Pharmacodynamics, Ecology, FoodWeb, CommunityDynamics, SpatialEcology,
  ReactionDiffusion, Chemotaxis, PatternFormation, StochasticBiology,
  BirthDeathProcess, BranchingProcess, MarkovProcess, NetworkBiology,
  ComputationalBiology, ParameterEstimation, SensitivityAnalysis,
  Identifiability, StabilityAnalysis, BifurcationAnalysis
 ]
 EXCLUDE_AS_PRIMARY = [
  DescriptiveBiologyWithoutMathematicalModel,
  PureChemistryWithoutBiologicalSystemContext,
  GeneralStatisticsWithoutBiologicalInterpretation
 ]
 DISAMBIGUATION_RULE = "When equations, probability models, networks or computational structures represent living systems or biological processes, interpret the context as biological mathematics."
}

CORE_CONCEPTS {
 BiologicalMathematics = "Mathematical representation and analysis of living systems."
 BiologicalModel = "A mathematical abstraction of a biological process or system."
 PopulationDynamics = "Changes in population size, composition or structure through time."
 Epidemiology = "Mathematical study of disease transmission and population-level disease dynamics."
 EvolutionaryDynamics = "Mathematical study of biological traits and fitness through selection and inheritance."
 SystemsBiology = "Mathematical study of interacting biological components as integrated systems."
 BiochemicalKinetics = "Mathematical modelling of biochemical reaction rates."
 SpatialBiology = "Mathematical study of biological processes distributed across space."
 StochasticBiology = "Representation of biological randomness through stochastic processes."
 NetworkBiology = "Representation of biological entities and interactions as mathematical networks."
}

RELATIONS {
 BiologicalMathematics --is_branch_of--> AppliedMathematics
 BiologicalMathematics --models--> BiologicalSystem
 BiologicalModel --contains--> StateVariable
 BiologicalModel --contains--> BiologicalParameter
 PopulationDynamics --uses--> DifferentialEquations
 LogisticGrowth --uses--> CarryingCapacity
 PredatorPreyModel --is_type_of--> PopulationDynamics
 Epidemiology --uses--> CompartmentalModel
 SIRModel --is_type_of--> CompartmentalModel
 BasicReproductionNumber --characterises--> EpidemicPotential
 EvolutionaryDynamics --depends_on--> Fitness
 ReplicatorEquation --models--> FrequencyDependentSelection
 PopulationGenetics --tracks--> AlleleFrequency
 SystemsBiology --uses--> NetworkBiology
 MichaelisMentenKinetics --models--> EnzymeKinetics
 ReactionDiffusion --models--> SpatialBiology
 Chemotaxis --models--> DirectedBiologicalMovement
 BirthDeathProcess --is_type_of--> StochasticBiology
 ParameterEstimation --calibrates--> BiologicalModel
 SensitivityAnalysis --assesses--> ParameterInfluence
 StabilityAnalysis --assesses--> BiologicalEquilibrium
}

OPERATORS {
 EXPONENTIAL_GROWTH(N,r) := dN/dt=r*N
 LOGISTIC_GROWTH(N,r,K) := dN/dt=r*N*(1-N/K)
 LOTKA_VOLTERRA_PREY(x,alpha,beta,y) := dx/dt=alpha*x-beta*x*y
 LOTKA_VOLTERRA_PREDATOR(y,delta,gamma,x) := dy/dt=delta*x*y-gamma*y
 SIR_S(S,beta,I,N) := dS/dt=-beta*S*I/N
 SIR_I(S,I,beta,gamma,N) := dI/dt=beta*S*I/N-gamma*I
 SIR_R(I,gamma) := dR/dt=gamma*I
 BASIC_REPRODUCTION_NUMBER(beta,gamma) := R0=beta/gamma under_basic_SIR_assumptions
 REPLICATOR_EQUATION(x_i,f_i,fbar) := dx_i/dt=x_i*(f_i-fbar)
 HARDY_WEINBERG(p,q) := frequencies=[p^2,2*p*q,q^2]
 MICHAELIS_MENTEN(S,Vmax,Km) := v=Vmax*S/(Km+S)
 REACTION_DIFFUSION(u,D,f) := partial u/partial t=D*nabla^2u+f(u)
}

AXIOMS {
 A1 = "Biological models must define the biological meaning of mathematical state variables."
 A2 = "Model parameters require biological interpretation and units."
 A3 = "Biological conclusions depend on model assumptions and scale."
 A4 = "Deterministic and stochastic biological models must not be conflated."
 A5 = "Population-level models may not directly represent individual-level mechanisms."
 A6 = "Spatial biological systems require explicit geometry, movement or diffusion assumptions."
 A7 = "Parameter identifiability is distinct from parameter sensitivity."
 A8 = "Biological models require validation against empirical or independently supported behaviour where possible."
}

REASONING {
 R1_CONTEXT = identify(biological_system,scale,state_variables,parameters,observations)
 R2_MODEL_CLASS = classify(population,epidemiological,evolutionary,genetic,biochemical,physiological,ecological,network,spatial,stochastic)
 R3_FORMULATION = construct(equations,transition_rules,boundary_conditions,initial_conditions)
 R4_EQUILIBRIUM = solve(steady_states,fixed_points)
 R5_STABILITY = analyse(local_stability,eigenvalues,thresholds)
 R6_DYNAMICS = simulate(time_evolution,oscillation,bifurcation,pattern_formation)
 R7_STOCHASTIC = analyse(distributions,transition_probabilities,extinction_or_fixation)
 R8_CALIBRATION = estimate(parameters,uncertainty,identifiability)
 R9_SENSITIVITY = quantify(parameter_and_initial_condition_influence)
 R10_VALIDATION = compare(model_predictions,biological_constraints,data,known_behaviour)
}

CONSTRAINTS {
 C1 = "Biological parameters should not be interpreted outside their calibrated context without justification."
 C2 = "Continuous population models may be inappropriate for very small populations."
 C3 = "Mass-action assumptions may fail in spatially structured or low-copy-number biochemical systems."
 C4 = "Basic reproduction numbers depend on model structure."
 C5 = "Equilibrium stability does not guarantee global stability."
 C6 = "Deterministic averages can hide extinction or rare-event behaviour."
 C7 = "Correlation between biological variables does not establish mechanistic causation."
 C8 = "Model complexity can exceed parameter identifiability."
 C9 = "Spatial discretisation can alter pattern-forming behaviour if resolution is inadequate."
 C10 = "A good fit to observed data does not uniquely validate a biological mechanism."
}

SEMANTIC_EQUIVALENCE {
 "BiologicalMathematics" <=> "MathematicalBiology"
 "PopulationDynamics" <=> "MathematicalPopulationDynamics"
 "BiochemicalKinetics" <=> "BiologicalReactionKinetics"
 "NetworkBiology" <=> "BiologicalNetworkMathematics"
 "StochasticBiology" <=> "StochasticMathematicalBiology"
}

VALIDATION {
 V1 = "Biological system and scale are explicit."
 V2 = "State variables have biological meanings and units."
 V3 = "Parameters and assumptions are documented."
 V4 = "Initial and boundary conditions are explicit where required."
 V5 = "Equilibria and stability are checked where relevant."
 V6 = "Parameter identifiability is assessed where calibration is performed."
 V7 = "Sensitivity and uncertainty are evaluated."
 V8 = "Stochastic simulation convergence is assessed where applicable."
 V9 = "Spatial and temporal numerical resolution are validated."
 V10 = "Model outputs are compared with biological evidence or constraints."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 BIOLOGICAL_MATHEMATICS_DEFINITION = 0.99
 MODEL_CLASSIFICATION_REASONING = 0.99
 POPULATION_REASONING = 0.99
 EPIDEMIOLOGY_REASONING = 0.98
 EVOLUTIONARY_REASONING = 0.98
 BIOCHEMICAL_REASONING = 0.98
 SPATIAL_REASONING = 0.98
 STOCHASTIC_REASONING = 0.98
 CALIBRATION_REASONING = 0.97
 POLICY = "Reduce confidence when biological scale, state variables, parameter meanings, mechanisms, initial conditions, spatial structure, data quality or validation evidence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, BiologicalMathematicsReasoning,
  PopulationDynamicsReasoning, EpidemiologicalReasoning, EvolutionaryReasoning,
  PopulationGeneticsReasoning, SystemsBiologyReasoning, BiochemicalKineticsReasoning,
  NeuralDynamicsReasoning, PhysiologicalReasoning, EcologicalReasoning,
  SpatialBiologyReasoning, ReactionDiffusionReasoning, StochasticBiologyReasoning,
  NetworkBiologyReasoning, ParameterEstimationReasoning, SensitivityReasoning,
  StabilityReasoning, KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "biological mathematics" -> BiologicalMathematics
  "mathematical biology" -> BiologicalMathematics
  "population dynamics" -> PopulationDynamics
  "mathematical epidemiology" -> Epidemiology
  "systems biology" -> SystemsBiology
  "biochemical kinetics" -> BiochemicalKinetics
  "network biology" -> NetworkBiology
  "stochastic biology" -> StochasticBiology
 }
}

EXECUTION_RULES {
 E1 = "If a living system is represented mathematically, activate BiologicalMathematics."
 E2 = "If population size changes through time, route to PopulationDynamics."
 E3 = "If infectious transmission is central, route to Epidemiology."
 E4 = "If selection, inheritance or allele frequencies are central, route to evolutionary or population-genetics mathematics."
 E5 = "If biochemical reaction rates are central, route to BiochemicalKinetics."
 E6 = "If biological interactions form a graph, route to NetworkBiology."
 E7 = "If spatial movement or diffusion is material, route to SpatialBiology or ReactionDiffusion."
 E8 = "If random events materially affect outcomes, route to StochasticBiology."
 E9 = "If model parameters are inferred from data, require identifiability, uncertainty and validation checks."
}

RECURSION {
 ENTRY = BiologicalMathematics
 EXPAND(BiologicalMathematics) -> [
  BiologicalSystem, BiologicalModel, PopulationDynamics, PopulationGrowth,
  ExponentialGrowth, LogisticGrowth, CarryingCapacity, Competition, Predation,
  PredatorPreyModel, LotkaVolterraModel, Epidemiology, CompartmentalModel,
  SIRModel, SEIRModel, BasicReproductionNumber, EndemicEquilibrium,
  EvolutionaryDynamics, NaturalSelection, Fitness, ReplicatorEquation,
  PopulationGenetics, AlleleFrequency, HardyWeinbergEquilibrium, GeneticDrift,
  Mutation, Migration, SystemsBiology, BiochemicalKinetics, EnzymeKinetics,
  MichaelisMentenKinetics, GeneRegulatoryNetwork, ProteinInteractionNetwork,
  MetabolicNetwork, NeuralDynamics, PhysiologicalModel, Pharmacokinetics,
  Pharmacodynamics, Ecology, FoodWeb, CommunityDynamics, SpatialEcology,
  ReactionDiffusion, Chemotaxis, PatternFormation, StochasticBiology,
  BirthDeathProcess, BranchingProcess, MarkovProcess, NetworkBiology,
  ComputationalBiology, ParameterEstimation, SensitivityAnalysis,
  Identifiability, StabilityAnalysis, BifurcationAnalysis
 ]
 STOP_CONDITION = "Stop when biological scale, mathematical model, mechanisms, parameters, dynamics, uncertainty and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"
 RELATED_ONTOLOGIES = [
  "Applied Mathematics Ontologies",
  "Differential Equation Ontologies",
  "Dynamical Systems Ontologies",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies",
  "Graph Theory Ontologies",
  "Optimisation Ontologies",
  "Computational Mathematics Ontologies",
  "Biological Ontologies"
 ]
 ROUTE_RULES = [
  "Biological Mathematics -> Applied Mathematics for parent semantics",
  "Biological Mathematics -> Differential Equations for continuous biological dynamics",
  "Biological Mathematics -> Dynamical Systems for equilibria, stability and bifurcation",
  "Biological Mathematics -> Probability & Statistics for uncertainty and inference",
  "Biological Mathematics -> Stochastic Processes for random biological dynamics",
  "Biological Mathematics -> Graph Theory for biological networks",
  "Biological Mathematics -> Optimisation for parameter fitting",
  "Biological Mathematics -> Computational Mathematics for numerical simulation",
  "Biological Mathematics -> Biological Ontologies for domain semantics"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible biological models, scales or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve biological-state semantics.",
  "Preserve scale semantics.",
  "Preserve deterministic-versus-stochastic distinction.",
  "Preserve mechanism-versus-observation distinction.",
  "Preserve spatial-versus-nonspatial distinction.",
  "Preserve parameter-identifiability semantics.",
  "Preserve validation semantics.",
  "Preserve machine-readable biological-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Biological mathematics applies mathematical models to living systems across molecular, cellular, organismal, population and ecological scales."
 POPULATION_SUMMARY = "Population models describe growth, competition, predation and population interactions."
 EPIDEMIOLOGY_SUMMARY = "Epidemiological models represent disease transmission, compartment transitions and threshold behaviour."
 EVOLUTION_SUMMARY = "Evolutionary and genetic mathematics model selection, fitness, inheritance, mutation and allele-frequency change."
 SYSTEMS_SUMMARY = "Systems biology models interacting genes, proteins, metabolites and biochemical reactions."
 SPATIAL_SUMMARY = "Reaction-diffusion, chemotaxis and spatial ecological models represent biological processes across space."
 STOCHASTIC_SUMMARY = "Stochastic biological models represent demographic, molecular and environmental randomness."
 VALIDATION_SUMMARY = "Biological models should be checked for mechanistic meaning, parameter identifiability, sensitivity, uncertainty and empirical consistency."
}

FLOW {
 INPUT
 -> detect_biological_mathematics_context
 -> identify_biological_system_scale_state_variables_parameters_data_and_questions
 -> classify_population_epidemiological_evolutionary_genetic_biochemical_physiological_ecological_network_spatial_or_stochastic_model
 -> formulate_equations_transition_rules_initial_conditions_and_boundary_conditions
 -> solve_or_simulate_dynamics
 -> analyse_equilibria_stability_thresholds_bifurcations_or_patterns
 -> if_stochastic_compute_distributions_extinction_fixation_or_uncertainty
 -> estimate_parameters_and_assess_identifiability
 -> perform_sensitivity_and_uncertainty_analysis
 -> validate_against_biological_constraints_and_data
 -> route_to_applied_mathematics_differential_equations_dynamical_systems_probability_statistics_stochastic_processes_graph_theory_optimisation_computational_mathematics_or_biological_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Biological Mathematics layer defines mathematical modelling of living systems. It supports population growth and interactions, predator-prey dynamics, epidemiological compartment models and reproduction thresholds, evolutionary dynamics and population genetics, systems biology, enzyme and biochemical kinetics, gene, protein and metabolic networks, neural and physiological models, pharmacokinetics and pharmacodynamics, ecology and food webs, spatial ecology, reaction-diffusion, chemotaxis and pattern formation, stochastic birth-death and branching processes, network biology, computational biology, parameter estimation, sensitivity, identifiability, stability and bifurcation analysis."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-BIOLOGICAL-MATHEMATICS-V1"
}

}

END_FLOW